cdfgam {lmomco} | R Documentation

## Cumulative Distribution Function of the Gamma Distribution

### Description

This function computes the cumulative probability or nonexceedance probability of the Gamma distribution given parameters (α and β) of the distribution computed by `pargam`. The cumulative distribution function of the distribution has no explicit form, but is expressed as an integral.

F(x) = frac{β^{-α}}{Γ(α)}int_0^x t^{α - 1} e^{-t/β} mbox{d}F mbox{,}

where F(x) is the nonexceedance probability for the quantile x. The parameters have the following interpretation in the R syntax; α is a shape parameter and β is a scale parameter.

### Usage

    cdfgam(x, para)

### Arguments

`x` | A real value.
`para` | The parameters from `pargam` or similar.

### Value

Nonexceedance probability (F) for x.

### Author(s)

W.H. Asquith

### References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105–124.

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

### See Also

`quagam`, `pargam`

### Examples

      lmr <- lmom.ub(c(123,34,4,654,37,78))
      cdfgam(50,pargam(lmr))

      # A manual demonstration of a gamma parent
      G  <- vec2par(c(0.6333,1.579),type='gam') # the parent
      F1 <- 0.25         # nonexceedance probability
      x  <- quagam(F1,G) # the lower quartile (F=0.25)
      a  <- 0.6333       # gamma parameter
      b  <- 1.579        # gamma parameter
      # compute the integral
      xf <- function(t,A,B) { t^(A-1)*exp(-t/B) }
      Q  <- integrate(xf,0,x,A=a,B=b)
      # finish the math
      F2 <- Q$val*b^(-a)/gamma(a)
      # check the result
      if(abs(F1-F2) < 1e-8) print("yes")

[Package lmomco version 0.92 Index]